... field the vanishing of the Euler-Poincar´e characteristics takes place when Eis the trivial local system.Using the fact that our initial perverse sheaf F was of geometric origin, the standard ... P(Bunn) and some local systemE of rank equal to that of E, the Euler-Poincar´e characteristics of the stalks of AvdE(F) vanish.When we are working in the -adic situation over a finite ... explained in the sequel, the vanishing of the functor AvdEis analogous to the condition that the Rankin-Selberg convolution of E, viewedas an m-dimensional Galois representation, and an automorphic...